some problems

Sector OAB is a quarter of a circle of radius 3. A circle is drawn inside this sector, tangent at three points as shown. What is the radius of the inscribed circle? Express your answer in simplest radical form.

Re: some problems

Q1 correct. Well done!

derp how did i not see that?!?!?!

It can happen to us all. I nearly messed up an exam because I hadn't noticed that the length of a radius on one side of a circle was the same size as the radius on the other side of the same circle. doh!

Q2 Call the point where XR and YS cross, point Z.

Triangles ZXY and ZRS are similar. This is because they have a common angle at Z and XY is parallel to SR.

So If you call the height of ZXY k then ZSR has height 4k (because the ratio of sides is 3:12)

So the height (distance) between XY and SR is 5k.

Now look at triangles ZXY and ZPQ. These are similar in the ratio k:6k so now you can calculate PQ.

Q3. I have made a diagram and labelled the points.

PVU is 90 as PV is a tangent

PWS is 90 as WX is the distance you are required to find.

So triangles PVU and PWS are similar.

So you can use the ratio of sides to work out WS, and then WX is easy.

Hope that helps,

Bob

T is the centre of a another circle, diameter PU that goes through V. I thought I needed this circle too, but, it turns out I don't. Interesting though.

Last edited by bob bundy (2014-08-10 05:27:31)

Children are not defined by school ...........The FonzYou cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

Re: some problems

thanks, i got both of them.

Let ABCD be a square, and let M and N be the midpoints of

and , respectively. Find .

Triangle ABC has side lengths AB = 9, AC = 10, and BC = 17. Let X be the intersection of the angle bisector of \angle A with side \overline{BC}, and let Y be the foot of the perpendicular from X to side \overline{AC}. Compute the length of \overline{XY}.

Equilateral triangle ABC and a circle with center O are constructed such that \overline{BC} is a chord of the circle and point A is the circumcenter of \triangle BCO in its interior. If the area of circle with center O is 48\pi, then what is the area of triangle ABC?

In a triangle ABC, take point D on \overline{BC} such that DB = 14, DA = 13, DC = 4, and the circumcircles of triangles ADB and ADC have the same radius. Find the area of triangle ABC.

Let

denote the circular region bounded by x^2 + y^2 = 36. The lines x = 4 and y = 3 partition into four regions . Letdenote the area of region If then compute

Re: some problems

Hello cooljackiec,I've noticed that you have been posting many AoPS problems along with their official diagrams. Please remove them as they constitute plagiarism and cause a permanent record that students who are attempting to solve the problems will see. Thank you.